Classical Probability Theories
In this chapter we present a brief introductions to Kolmogorov’s measure-theoretic and von Mises’ frequency probability theories. These are classical probability theories. A rather common opinion is that classical probability can not describe the probabilistic structure of quantum mechanics. Such an opinion is based on belief that classical probability can not reproduce the main distinguishing probabilistic features of quantum formalism, e.g., interference of probabilities and Born’s rule. It is also believed that the violation of Bell’s inequality can not be explained in the framework of classical probability theory. The aim of this book is to show that such a viewpoint to the relation between classical and quantum probabilities is not justified. We shall see that the mathematical formalisms of Kolmogorov’s measure-theoretic [Kolmogoroff, Grundbegriffe der Wahrscheinlichkeitsrechnung, 1933; Shiryaev in Ann. Probab. 17:866–944, 1989; Shiryaev, Probability, 1984] and Mises’ frequency [Von Mises in Math. Z. 5:52–99, 1919; Von Mises, Probability, Statistics and Truth, 1957; Von Mises, The Mathematical Theory of Probability and Statistics, 1964] models can be represented in the QL way. However, these models should be interpreted in the contextual framework.
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